On Covering paths with 3 Dimensional Random Walk

TL;DR

This paper investigates the probability that a 3D simple random walk covers a specific path, providing an upper bound that decays slightly slower than exponential as the path length increases.

Contribution

It extends previous results by deriving a new upper bound for the covering probability in three dimensions, where earlier techniques for higher dimensions do not apply.

Findings

Probability decays as N log^{-(1+ε)}(N) for 3D walks

Upper bound is weaker than exponential decay in 3D

Results differ from known behavior in higher dimensions

Abstract

In this paper we find an upper bound for the probability that a $3$ dimensional simple random walk covers each point in a nearest neighbor path connecting 0 and the boundary of an $L_1$ ball of radius $N$. For $d\ge 4$, it has been shown in [5] that such probability decays exponentially with respect to $N$. For $d=3$, however, the same technique does not apply, and in this paper we obtain a slightly weaker upper bound: $\forall \varepsilon>0,\exists c_\varepsilon>0,$ $$P\left({\rm Trace}(\mathcal{P})\subseteq {\rm Trace}\big(\{X_n\}_{n=0}^\infty\big) \right)\le \exp\left(-c_\varepsilon N\log^{-(1+\varepsilon)}(N)\right).$$

Equations

$$P\left({\rm Trace}(\mathcal{P})\subseteq{\rm Trace}\big{(}\{X_{n}\}_{n=0}^{\infty}\big{)}\right)\leq\exp\left(-c_{\varepsilon}N\log^{-(1+{\varepsilon})}(N)\right).$$

$$\mathcal{P}=\big{(}P_{0},P_{1},\cdots,P_{K}\big{)}$$

$$\mathrm{Trace}(\mathcal{P})\subseteq \mathrm{Trace}(X_{d,0},X_{d,1},\cdots ):=\{x\in {\mathbb{Z}}^d,\exists n{X}_{d,n}=x\}.$$

$$P\big{(}{\rm Trace}(\mathcal{P})\in{\rm Trace}(X_{d,0},\cdots,X_{d,L})\big{)}\leq P\big{(}\overset{\nearrow}{\mathcal{P}}\in{\rm Trace}(X_{d,0},\cdots,X_{d,L})\big{)}$$

$$\overset{\nearrow}{\mathcal{P}}=\Big{(}{\rm arc}_{1}[0:d-1],{\rm arc}_{2}[0:d-1],\cdots,{\rm arc}_{[N/d]}[0:d-1],{\rm arc}_{[N/d]+1}[0:N-d[N/d]]\Big{)},$$

$${\mathrm{arc}}_1[0:d-1]=\left(0,e_1,e_1+e_2,\cdots ,\sum _{i=1}^{d-1}{e}_i\right)$$

$${\hat{X}}_{d-1,n}=\left(X_{d,n}^1-X_{d,n}^2,X_{d,n}^2-X_{d,n}^3,\cdots ,X_{d,n}^{d-1}-X_{d,n}^d\right)$$

$$P\left({\rm Trace}(\mathcal{P})\subseteq{\rm Trace}\big{(}\{X_{d,n}\}_{n=0}^{\infty}\big{)}\right)\leq P_{d}^{[N/d]}.$$

$${\mathcal{D}}_{\infty }=\{(0,0,0),(1,1,1),\cdots \}$$

$$P\big{(}\overset{\nearrow}{\mathcal{P}}\in{\rm Trace}(X_{3,0},X_{3,1},\cdots)\big{)}\leq\exp\left(-c\Big{|}\mathcal{\tilde{D}}_{\infty}\cap\overset{\nearrow}{\mathcal{P}}\Big{|}\right).$$

$$P\left({\rm Trace}(\mathcal{P})\subseteq{\rm Trace}\big{(}\{X_{3,n}\}_{n=0}^{\infty}\big{)}\right)\leq\exp\left(-c_{\varepsilon}N\log^{-(1+{\varepsilon})}(N)\right).$$

$$P\left({\rm Trace}(\mathcal{P})\subseteq{\rm Trace}\big{(}\{X_{3,n}\}_{n=0}^{\infty}\big{)}\right)\leq\exp\left(-cN\right).$$

$$\overset{\nearrow}{\mathcal{P}}=\Big{(}{\rm arc}_{1}[0:d-1],{\rm arc}_{2}[0:d-1],\cdots,{\rm arc}_{[N/d]}[0:d-1],{\rm arc}_{[N/d]+1}[0:N-d[N/d]]\Big{)}$$

$${\mathcal{D}}_{[N/3]}=\{(0,0,0),(1,1,1),\cdots ,([N/3],[N/3],[N/3])\}$$

$$n_k=\lceil \sum _{i=1}^k{\log }^{1+\epsilon }(i)\rceil \in \mathbb{Z},$$

$${\tilde{\mathcal{D}}}_{\infty }=\{(n_k,n_k,n_k){\}}_{k=1}^{\infty }\subset {\mathcal{D}}_{\infty }.$$

$$n_{k_2}-n_{k_1}$$

$$\ge \sum _{i=k_1+1}^{k_2}{\log }^{1+\epsilon }(i)-1.$$

$$n_{k_2}-n_{k_1}\ge \frac{1}{2}{\int }_{k_1}^{k_2}{\log }^{1+\epsilon }(x)dx.$$

$${\tilde{\mathcal{D}}}_N={\tilde{\mathcal{D}}}_{\infty }\cap {\mathcal{D}}_N$$

$$C_N=\left|{\tilde{\mathcal{D}}}_N\right|=\sup \{k:n_k\le N\}.$$

$$C_N=\sup \left\{k:\sum _{i=1}^k{\log }^{1+\epsilon }(i)\le N\right\}=\inf \left\{k:\sum _{i=1}^k{\log }^{1+\epsilon }(i)>N\right\}-1.$$

$$C_N\in \left(2^{-1-\epsilon }N{\log }^{-1-\epsilon }(N),C_{\epsilon }N{\log }^{-1-\epsilon }(N)\right)$$

$$\sum _{i=1}^k{\log }^{1+\epsilon }(i)>N$$

$$\displaystyle\sum_{i=1}^{k}\log^{1+{\varepsilon}}(i)\geq\int_{1}^{k}\log^{1+{\varepsilon}}(x)dx\geq\frac{1}{2^{1+{\varepsilon}}}\big{(}k-k^{1/2}\big{)}\log^{1+{\varepsilon}}(k),$$

$$\sum _{i=1}^{K_N}{\log }^{1+\epsilon }(i)$$

$$\ge \frac{1}{2^{1+\epsilon }}\cdot K_N\cdot \frac{K_N-K_N^{1/2}}{K_N}\cdot {\log }^{1+\epsilon }\left(2^{2+\epsilon }N/{\log }^{1+\epsilon }(N)\right)$$

$$\ge 2N\cdot \frac{K_N-K_N^{1/2}}{K_N}\cdot \frac{{\log }^{1+\epsilon }\left(2^{2+\epsilon }N/{\log }^{1+\epsilon }(N)\right)}{{\log }^{1+\epsilon }(N)}.$$

$$\underset{N\to \infty }{\lim }\frac{{\log }^{1+\epsilon }\left({\log }^{1+\epsilon }(N)\right)}{{\log }^{1+\epsilon }(N)}=\underset{N\to \infty }{\lim }(1+\epsilon {)}^{1+\epsilon }{\left[\frac{\log (\log (N))}{\log (N)}\right]}^{1+\epsilon }=0,$$

$$\sum _{i=1}^{K_N}{\log }^{1+\epsilon }(i)\ge 2N\cdot \frac{K_N-K_N^{1/2}}{K_N}\cdot \frac{{\log }^{1+\epsilon }\left(2^{2+\epsilon }N/{\log }^{1+\epsilon }(N)\right)}{{\log }^{1+\epsilon }(N)}>N$$

$$\sum _{i=1}^k{\log }^{1+\epsilon }(i)\le {\int }_1^{k+1}{\log }^{1+\epsilon }(x)dx\le k{\log }^{1+\epsilon }(k+1).$$

$$\sum _{i=1}^k{\log }^{1+\epsilon }(i)\le k{\log }^{1+\epsilon }(k+1)\le 2^{-1-\epsilon }N\frac{{\log }^{1+\epsilon }\left(2^{-1-\epsilon }N{\log }^{-1-\epsilon }(N)+1\right)}{{\log }^{1+\epsilon }(N)}<N.$$

Full text

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1. Introduction

In this paper, we study the probability that the trace of a nearest neighbor path in $\mathbb{Z}^{3}$ connecting 0 and the boundary of a $L_{1}$ ball of radius $N$ is completely covered by the trace of a $3$ dimensional simple random walk.

First, we review some results we proved in a recent paper for general $d$’s. For any integer $N\geq 1$, let $\partial B_{1}(0,N)$ be the boundary of the $L_{1}$ ball in $\mathbb{Z}^{d}$ with radius $N$. We say that a nearest neighbor path

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is connecting 0 and $\partial B_{1}(0,N)$ if $P_{0}=0$ and $\inf\{n:\|P_{n}\|_{1}=N\}=K$. And we say that a path $\mathcal{P}$ is covered by a $d$ dimensional random walk $\{X_{d,n}\}_{n=0}^{\infty}$ if

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In [5], we have shown that for any $d\geq 2$ such covering probability is maximized over all nearest neighbor paths connecting 0 and $\partial B_{1}(0,N)$ by the monotonic path that stays within distance one above/below the diagonal $x_{1}=x_{2}=\cdots=x_{d}$.

Theorem 1.1**.**

*(Theorem 1.4 in [5])

For each integers $L\geq N\geq 1$, let $\mathcal{P}$ be any nearest neighbor path in $\mathbb{Z}^{d}$ connecting 0 and $\partial B_{1}(0,N)$. Then*

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where

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and ${\rm arc}_{k}=(k-1)\sum_{i=1}^{d}e_{i}+{\rm arc}_{1}$.

Then noting that the probability of covering $\overset{\nearrow}{\mathcal{P}}$ is bounded above by the probability a simple random walk returns to the exact diagonal line for $[N/d]$ times, one can introduce the Markov process

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where $X_{d,n}^{i}$ is the $i$th coordinate of $X_{d,n}$ and see that $\{\hat{X}_{d-1,n}\}_{n=0}^{\infty}$ is another $d-1$ dimensional non simple random walk, which is transient when $d\geq 4$. Thus, we immediately have the following upper bound:

Theorem 1.2**.**

*(Theorem 1.5 in [5])

There is a $P_{d}\in(0,1)$ such that for any nearest neighbor path $\mathcal{P}=(P_{0},P_{1},\cdots,P_{K})$ connecting 0 and $\partial B_{1}(0,N)$ and $\{X_{d,n}\}_{n=0}^{\infty}$ which is a $d-$dimensional simple random walk starting at 0 with $d\geq 4$, we always have*

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Here $P_{d}$ equals to the probability that $\{X_{d,n}\}_{n=0}^{\infty}$ ever returns to the $d$ dimensional diagonal line.

Theorem 1.2 implies that for each fixed $d\geq 4$, the covering probability decays exponentially with respect to $N$.

For $d=3$, the same technique we had may not hold since now $\{\hat{X}_{2,n}\}_{n=0}^{\infty}$ is a recurrent 2 dimensional random walk, which means that $P_{d}=1$ and that the original 3 dimensional random walk will return to the diagonal line infinitely often. To overcome this issue, we note that although the diagonal line

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is recurrent, it is possible to find an infinite subset $\mathcal{\tilde{D}}_{\infty}$ that is transient. And if we can further show for this specific transient subset we found, the returning probability is uniformly bounded away from 1 (which is not generally true for all transient subsets, as is shown in Counterexample 1 in Section 3), then we are able to show

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With this approach, we have the following theorem

Theorem 1.3**.**

For each ${\varepsilon}>0$, there is a $c_{\varepsilon}\in(0,\infty)$ such that for any $N\geq 2$ and any nearest neighbor path $\mathcal{P}=(P_{0},P_{1},\cdots,P_{K})\subset\mathbb{Z}^{3}$ connecting 0 and $\partial B_{1}(0,N)$, we have

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Note that the upper bound in Theorem 1.3 seems to be non-sharp. The reason is that we did not fully use the geometric property of path $\overset{\nearrow}{\mathcal{P}}$ to minimize the covering probability. I.e., although we require our simple random walk to visit the transient subset for $O(N\log^{-1-{\varepsilon}}(N))$ times, those returns may be not enough to cover every point in $\mathcal{\tilde{D}}_{\infty}\cap\overset{\nearrow}{\mathcal{P}}$. We conjecture that the actual decay rate is also exponential for $d=3$. Numerical simulations seem to support this as is shown in Section 5.

Conjecture 1.4**.**

There is a $c\in(0,\infty)$ such that for any $N\geq 2$ and any nearest neighbor path $\mathcal{P}=(P_{0},P_{1},\cdots,P_{K})\subset\mathbb{Z}^{3}$ connecting 0 and $\partial B_{1}(0,N)$, we always have

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The structure of this paper is as follows: In Section 2, we construct the infinite subset $\mathcal{\tilde{D}}_{\infty}$ of the diagonal line, calculate its density, and show it is transient. In Section 3, we show the returning probability of $\mathcal{\tilde{D}}_{\infty}$ is uniformly bounded away from 1, no matter where on $\mathcal{\tilde{D}}_{\infty}$ the random walk starts from. With these results, in Section 4, we finish the proof of Theorem 1.3. Numerical simulations are given in Section 5 showing possible non tightness of our result.

2. Infinite Transient Subset of the Diagonal

Without loss of generality we can concentrate on the proof of Theorem 1.3 for sufficiently large $N$. Recall that

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is the path connection 0 and $B_{1}(0,N)$ that maximizes the covering probability. When $d=3$, let

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be the points in $\overset{\nearrow}{\mathcal{P}}$ that lie exactly on the diagonal. Although it is clear that for simple random walk $\{X_{3,n}\}_{n=0}^{\infty}$ starting at 0, $\mathcal{D}_{\infty}$ is a recurrent set, following a similar construction to Spitzer [6, Chapter 6.26], we find a transient infinite subset of $\mathcal{D}_{\infty}$ as follows: for $n_{1}=0$, $n_{2}=\lceil\log^{1+{\varepsilon}}(2)\rceil=1$, and for all $k\geq 3$

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define

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Since $\log^{1+{\varepsilon}}(k)>1$ for all $k\geq 3$, it is easy to see that $\{n_{k}\}_{k=1}^{\infty}$ is a monotonically increasing sequence. Moreover, for each $1\leq k_{1}<k_{2}<\infty$,

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This implies that for all $k_{2}\geq 8$ and $1\leq k_{1}<k_{2}$,

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For any $N\in\mathbb{Z}$, define

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and

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Recalling the definition of $n_{k}$ in (2.1), we also equivalently have

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Lemma 2.1**.**

For any ${\varepsilon}>0$, there is a constant $C_{\varepsilon}<\infty$ such that

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for all $N\geq 2$.

Proof.

Noting that for any $k$ such that

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we must have that $k>C_{N}$, and that

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for $K_{N}=\left\lceil 2^{2+{\varepsilon}}N/\log^{1+{\varepsilon}}(N)\right\rceil$, we have by (2.3)

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Noting that $K_{N}\to\infty$ as $N\to\infty$ and that

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for sufficiently large $N$

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which implies $C_{N}<K_{N}$ and finishes the proof of the upper bound. On the other hand, note that

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So for any $k\leq 2^{-1-{\varepsilon}}N\log^{-1-{\varepsilon}}(N)$,

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Thus the proof of Lemma 2.1 is complete. ∎

With Lemma 2.1, we next show that $\mathcal{\tilde{D}}_{\infty}$ is transient for 3 dimensional simple random walk:

Lemma 2.2**.**

For 3 dimensional simple random walk $\{X_{3,n}\}_{n=0}^{\infty}$, $\mathcal{\tilde{D}}_{\infty}$ is a transient subset.

Proof.

According to Wiener’s test (see Corollary 6.5.9 of [3]), it is sufficient to show that

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where $A_{k}=\mathcal{\tilde{D}}_{\infty}\cap\left[B_{2}(0,2^{k})\setminus B_{2}(0,2^{k-1})\right]$. Then according to the definition of capacity (see Section 6.5 of [3]), we have for all $k\geq 1$

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By Lemma 2.1,

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Thus we have

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which implies that $\mathcal{\tilde{D}}_{\infty}$ is transient. ∎

3. Uniform Upper Bound on Returning Probability

Now we have $\mathcal{\tilde{D}}_{\infty}$ is transient, i.e.,

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which immediately implies that there must be some $\bar{x}\in\mathbb{Z}^{3}\setminus\mathcal{\tilde{D}}_{\infty}$ such that

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where $T_{\mathcal{\tilde{D}}_{\infty}}$ is the first time a simple random walk visits $\mathcal{\tilde{D}}_{\infty}$, and $P_{x}(\cdot)$ is the distribution of the simple random walk condition on it starting at $x$. Then note that $\mathcal{\tilde{D}}_{\infty}$ is a subset of the diagonal line, which implies $\mathcal{\tilde{D}}_{\infty}$ has no interior point while $\mathbb{Z}^{3}\setminus\mathcal{\tilde{D}}_{\infty}$ is connected. Thus for any $x_{k}\in\mathcal{\tilde{D}}_{\infty}$, there exists a nearest neighbor path

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with $y_{0}=x_{k}$, $y_{m}=\bar{x}$ while $y_{i}\in\mathbb{Z}^{3}\setminus\mathcal{\tilde{D}}_{\infty}$, for all $i=1,2,\cdots,m-1$. Combining this with the fact that

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for all $x\in\mathbb{Z}^{3}\setminus\mathcal{\tilde{D}}_{\infty}$, we have

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for all $i\geq 1$, which in turns implies that

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for all $k$, where $\bar{T}_{\mathcal{\tilde{D}}_{\infty}}$ is the first returning time, i.e. the stopping time a simple random walk first visit $\mathcal{\tilde{D}}_{\infty}$ after its first step.

However, in order to use the transient set $\mathcal{\tilde{D}}_{\infty}$ as if it is just like one point in a transient random walk, (3.2) is not enough. We need to show that starting from each point $x_{k}=(n_{k},n_{k},n_{k})\in\mathcal{\tilde{D}}_{\infty}$, the probability $P_{x_{k}}(\bar{T}_{\mathcal{\tilde{D}}_{\infty}}<\infty)$ is uniformly bounded away from 1. And this is not generally true for all transient subsets $A$. First of all, when $A$ has interior points, the returning probability of those points are certainly one. And even if $A$ has no interior point and $\mathbb{Z}^{3}\setminus A$ is connected, we have the following counter example:

Counterexample 1: Consider subsets

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and

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where the 2 dimensional projection of $A$ is illustrated in Figure 1 (the distances between $A_{k}$’s are not exact in the figure):

Using Wiener’s test, it is easy to see $A$ is a transient subset. However, for points $a_{k}=(2^{k},0,0)\in A$, $k\geq 1$, in order to have a simple random walk starting at $a_{k}$ never returns to $A$, we must have the first $k$ steps of the random walk be along the $z-$coordinate. Thus

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which implies that

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Remark 3.1*.*

It would be interesting to characterize uniformly transient sets i.e. sets with uniformly bounded return probabilities.

Fortunately, for the specific transient subset $\mathcal{\tilde{D}}_{\infty}$, since it becomes more and more sparse as $x\to\infty$, we can still have:

Lemma 3.2**.**

For any ${\varepsilon}>0$, there is a $c_{{\varepsilon},1}>0$ such that

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Proof.

With (3.2) showing all returning probabilities are strictly less than 1, it is sufficient for us to show that

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Actually, here we prove a stronger statement

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Note that for each $k$

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and that

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It suffices for us to show that

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and that

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To show (3.6) and (3.7), we first note the well known result that there is a $C<\infty$ such that for any $x\not=y\in\mathbb{Z}^{3}$,

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First, to show (3.6) we have according to (2.2), for any $k\geq 8$

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Thus it is again sufficient to show that

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Note that

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For each $k\geq 8$ and $i\leq[k^{1/2}]$, we have

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Thus

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Then for each $k\geq 8$ and $i\in\left[\left\lceil k^{1/2}\right\rceil,k-1\right]$,

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Thus

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Noting that

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one can immediately have

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Combining (3.9), (3.11) and (3.13), we obtain (3.6).

Then, to show (3.7) we have according to (2.2), for any $k\geq 8$

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Thus it is again sufficient to show that

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Now for each $k$ we separate the infinite summation in (3.15) as

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For its first term we use similar calculation as in (3.12) and have

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And since

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we have

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At last for the second term in (3.16), we have for each $k\geq 8$ and $i\geq k^{2}+1$,

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Thus

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Finally, noting that

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we have the tail term

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as $k\to\infty$. Thus combining (3.15)- (3.20), we have shown (3.7) and thus finished the proof of this lemma. ∎

Proof of Theorem 1.3.

With Lemma 3.2, and recalling that

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and

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we can define the stopping times $\bar{T}_{\mathcal{\tilde{D}}_{[N/3]},0}=0$,

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and for all $k\geq 2$

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Then by Lemma 3.2, one can immediately see that for any $k\geq 0$

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and thus

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By Lemma 2.1 we have

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for all $N\geq 4$. Thus combining (3.21) and (3.22)

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where $c_{\varepsilon}=-\frac{2^{-{\varepsilon}-2}}{3}\log(1-c_{{\varepsilon},1})$. And the proof of Theorem 1.3 is complete. ∎

4. Discussions

In Conjecture 1.4, we conjecture that the cover probability should have exponential decay just as the $d\geq 4$ case. This conjecture seems to be supported by the following preliminary simulation which shows the log-plot of probabilities that the first $5000$ steps of a 3 dimensional simple random walk starting at 0 cover $\mathcal{D}_{i}=\{(0,0,0),(1,1,1),\cdots,(i,i,i)\}$ for $i=1,2,\cdots,9$.

The simulation result above seems to indicate that after taking logarithm, the covering probability decays almost exactly as a linear function, which implies the exponential decay we predicted, indicating that the upper bound we found in Theorem 1.3 is not sharp. For $N=9$, if Theorem 1.3 were sharp and there were a correction greater than $\log(N)$ in the exact decaying rate, then in the log-plot, it would cause the point to be $\log\log(9)\geq 0.787$ above the line. This is not seen in the simulation. However, the simulation above does not rule out the possibility that there is correction of a smaller order than $\log(N)$, since it could be so small for the initial 9 $i$’s and thus has not be seen significantly yet in the current simulation.

Another possible approach towards a sharp asymptotic is noting that although $\{\hat{X}_{2,n}\}_{n=0}^{\infty}$ is recurrent and will return to 0 with probability 1, the expected time between each two successive returns is $\infty$. Moreover, in order to cover $\overset{\nearrow}{\mathcal{P}}$, only those returns to diagonal before that $\{X_{3,n}\}_{n=0}^{\infty}$ has left $B_{2}(0,N)\supset B_{1}(0,N)$ forever could possibly help. This observation, together with the tail probability asymptotic estimations using local central limit theorem and techniques in [1] and [2] applied on the non simple random walk $\{\hat{X}_{2,n}\}_{n=0}^{\infty}$, and some large deviation argument, enable us to find a proper value of $T$ such that

• •

with high probability $\{X_{3,n}\}_{n=T}^{\infty}\cap B_{2}(0,N)=\O$,

• •

with high probability $\{\hat{X}_{2,n}\}_{n=0}^{T}$ will not return to 0 for $[N/3]$ times or more.

Right now this approach can only give us the following weaker upper bound (a detailed proof can be found in technical report [4]):

Proposition 4.1**.**

There are $c,C\in(0,\infty)$ such that for any nearest neighbor path $\mathcal{P}=(P_{0},P_{1},\cdots,P_{K})\subset\mathbb{Z}^{3}$ connecting 0 and $\partial B_{1}(0,N)$,

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However, this seemingly worse approach might have the potential to fully use the geometric property of path $\overset{\nearrow}{\mathcal{P}}$ to minimize the covering probability. Note that in order to cover $D_{[N/3]}$ we not only need $\{\hat{X}_{2,n}\}_{n=0}^{\infty}$ return to 0 for at least $[N/3]$ times before $\{X_{3,n}\}_{n=0}^{\infty}$ forever leaving $B_{2}(0,N)$, but also must have the locations of $X_{3,n}$ at such visits cover each point on the diagonal (let alone the request of covering the off diagonal points as well). I.e., define the stopping times $\tau_{l_{3},0}=0$

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and for all $i\geq 2$

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Define

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Noting that $\tau_{l_{3},i}<\infty$ for any $i$, and that $\{X_{3,n}\}_{n=0}^{\infty}$ is translation invariant, $\{Z_{3,n}\}_{n=0}^{\infty}$ is a well defined one dimensional random walk with infinite range. And we have

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Thus Conjecture 1.4 would follow from the techniques described above for Proposition 4.1 if the following conjecture is proved.

Conjecture 4.2**.**

There is a $c\in(0,\infty)$ such that for any $N\geq 2$

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